Question

Consider a binomial random variable with \(n=9\) and \(p=.3 .\) Let \(x\) be the number of successes in the sample. Evaluate the probabilities in Exercises \(7-10 .\) $$ P(2 \leq x \leq 4) $$

Short answer

Solution: The probability that the number of successes (x) ranges from 2 to 4 is approximately 0.7653.

Step-by-step solution

Identify the binomial probability formula

We'll use the binomial probability formula to find the probability for each value of x in the desired range [2, 4]: $$ P(x) = \binom{n}{x} p^x (1-p)^{n-x} $$ where \(n=9\), \(p=0.3\), and \(x\) takes on the values 2, 3, and 4.

Calculate probabilities for x = 2, 3, and 4

Use the binomial probability formula to calculate the probabilities for \(x=2\), \(x=3\), and \(x=4\). For x = 2: $$ P(x=2) = \binom{9}{2} (0.3)^2 (1-0.3)^{9-2} = 36(0.09)(0.7)^7 \approx 0.2668 $$ For x = 3: $$ P(x=3) = \binom{9}{3} (0.3)^3 (1-0.3)^{9-3} = 84(0.027)(0.7)^6 \approx 0.2984 $$ For x = 4: $$ P(x=4) = \binom{9}{4} (0.3)^4 (1-0.3)^{9-4} = 126(0.0081)(0.7)^5 \approx 0.2001 $$

Sum the probabilities

Now, sum the probabilities of \(x=2\), \(x=3\), and \(x=4\) to find the probability of \(2 \leq x \leq 4\): $$ P(2 \leq x \leq 4) = P(x=2) + P(x=3) + P(x=4) \approx 0.2668 + 0.2984 + 0.2001 \approx 0.7653 $$ So, the probability \(P(2 \leq x \leq 4)\) is approximately 0.7653.

Key concepts

Binomial Random Variable

A binomial random variable represents the number of successes in a sequence of independent trials of a binomial experiment. The key characteristics of such an experiment include having a fixed number of trials, each trial having only two possible outcomes (success or failure), the probability of success being the same for each trial, and each trial being independent of the others.

For example, flipping a coin several times, where we define 'success' as landing on heads, and 'failure' as landing on tails, can be described by a binomial random variable. In the case of the exercise with nine trials and probability of success as 0.3, we have a binomial random variable where 'success' could be drawing a certain colored ball from a bag, 'failure' not drawing that color, and repeating this 9 times with the probability of drawing the color being 30% for each draw.

Probability Distribution

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. For a binomial random variable, the probability distribution is known as a binomial distribution. It tells us how likely it is to achieve a certain number of 'successes' in a fixed number of trials.

In simpler terms, a probability distribution is like a chart that lists all the possible outcomes of a random process and the likelihood of each outcome happening. If you plotted these probabilities on a graph, you'd have a visual representation of the distribution. The binomial distribution, therefore, is quite specific: it details the probability of getting exactly 'x' successes in 'n' trials, which can be visualized as a series of bars, each bar's height showing the probability of a particular number of successes.

Binomial Theorem

The binomial theorem provides a quick way to expand an algebraic expression raised to a power, such as \( (a + b)^n \). It tells us that we can write this expression as the sum of terms of the form \( \binom{n}{k} a^{n-k} b^k \).

In the context of probability, the binomial theorem gives us a way to calculate the terms of the binomial expansion, which are used in the formulas for the binomial probability distribution. Each term of this expansion corresponds to the probability of having a certain number of successes in a series of trials within a binomial experiment. For example, in the exercise, the binomial theorem helps us calculate the probabilities of having exactly 2, 3, or 4 successes out of 9 trials, contributing to a clear understanding of binomial probabilities.